The General Power Formula
Logarithmic Functions
Exponential Functions
Trigonometric Functions
Trigonometric Transformation
Inverse Trigonometric Functions
Evaluate the following:
Example 4: $\displaystyle \int \sqrt{x^3 + 2} \,\, x^2 \, dx$
Example 5: $\displaystyle \int \dfrac{(3x^2 + 1) \, dx}{\root 3\of {(2x^3 + 2x + 1)^2}}$
Example 6: $\displaystyle \int (1 - 2x^2)^3 \, dx$
Evaluate the following integrals:
Example 1: $\displaystyle \int \dfrac{2x^3+5x^2-4}{x^2}dx$
Example 2: $\displaystyle \int (x^4 - 5x^2 - 6x)^4 (4x^3 - 10x - 6) \, dx$
Example 3: $\displaystyle \int (1 + y)y^{1/2} \, dy$
The definite integral of f(x) is the difference between two values of the integral of f(x) for two distinct values of the variable x. If the integral of f(x) dx = F(x) + C, the definite integral is denoted by the symbol
The quantity F(b) - F(a) is called the definite integral of f(x) between the limits a and b or simply the definite integral from a to b. It is called the definite integral because the result involves neither x nor the constant C and therefore has a definite value. The numbers a and b are called the limits of integration, a being the lower limit and b the upper limit.
In these formulas, u and v denote differentiable functions of some independent variable (say x) and a, n, and C are constants.
$$\displaystyle \int du = u + C$$
If F(x) is a function whose derivative F'(x) = f(x) on certain interval of the x-axis, then F(x) is called the anti-derivative of indefinite integral f(x). When we integrate the differential of a function we get that function plus an arbitrary constant. In symbols we write
Problem 648
For the cantilever beam loaded as shown in Fig. P-648, determine the deflection at a distance x from the support.
